Here, the subscript \(\tau \)  denotes the tangent component of a vector \({w}\) , i.e. \({w}_\tau ={w}-({w}\cdot \mathbf {n})\mathbf {n}\) , \(\nabla _\Gamma \)  is the tangential gradient on \(\Gamma \)  and \(\Delta _\Gamma \)  denotes the Laplace–Beltrami operator on \(\Gamma \)  (see, e.g., [1]} for differential operators on surfaces). The phenomenological parameters \(\beta \) , \(\kappa \)  and \(\gamma \)  are positive, and \(G\)  represents a surface double-well potential.
The surface dynamic of the concentration \(\psi \)  is described by () that is a convective Allen–Cahn type equation on the surface. It can be understood as an adaptation of the dynamic boundary condition (REF ) (with \(\delta =1)\)  to the situation of an additional volume-averaged velocity field. Notice that the boundary condition () (together with either () or ()) entails
that \(\int _\Omega \phi (t) \;\mathrm {d}x\)  is invariant (as for ()), whereas \(\int _\Gamma \psi (t) \;\mathrm {d}\Gamma \)  is not a conserved quantity.
The model H () endowed with () was firstly studied in [2]}. The authors proved the existence of global weak solutions assuming that both \(F\)  and \(G\)  are polynomial-like functions or \(F\)  is a singular potential and \(G\)  is polynomial function. In the former case, they also showed the convergence of any weak solution to a stationary state.
Later on, the global existence of weak solutions for () endowed with () has been achieved in [3]}. More recently, the compressible Navier–Stokes–Cahn–Hilliard system supplemented with () has been studied in [4]}, and the Navier–Stokes–Allen–Cahn and the Navier–Stokes–Voigt–Allen–Cahn systems with () have been analyzed in [5]}. For the stochastic model H with dynamic boundary conditions (), we mention the recent results in [6]}. It is worth pointing out that none of the aforementioned works established any uniqueness result for weak solutions in either two or three dimensions. In fact, the only uniqueness result established so far concerns quasi-strong solutions for the Navier–Stokes–Voigt–Allen–Cahn systems with () in [5]}.
